This repository collects several projects from my coursework in mathematics and statistics at the University of Michigan, Ann Arbor.
This is a coding project for STATS 506, where I implement simulations of St. Petersburg game and document the functions into a custom R package. The St. Petersburg game is a classical example where the payoff \[ X = 2^K,\quad \mathbb{P}(X = 2^k) = 2^{-k},\ k \ge 1 \] has infinite expectation, yet empirical averages behave very irregularly. Monte Carlo simulation of repeated plays of the game enable us to visualize and study the convergence behavior of sample mean and scaled averages \[ A_n = \frac{1}{n \log_2 n} \sum_{i=1}^n X_i. \] We also discussed the link of this problem to robust estimation.
This project is related to my coursework in MATH 440, MATH 656, and MATH 651. It focuses on time-dependent PDEs (e.g. the 1D Saint-Venant / shallow-water system) and compares:
Figure 1. Solutions of 1D Saint_Venant equations with PINNs.